Writing Equations of Lines Worksheets with Answer Key - Free Printable

Let’s solve each problem step by step.

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Problem 1:
Given:
- Line 1: 3y = 3x + 15
- Line 2: 2x - 2y = 2

We need to find the slope of each line. If slopes are equal → parallel.
If slopes multiply to -1 → perpendicular. Otherwise → none.

→ Convert both to slope-intercept form (y = mx + b)

Line 1:
3y = 3x + 15
Divide both sides by 3:
y = x + 5 → slope m₁ = 1

Line 2:
2x - 2y = 2
Subtract 2x from both sides:
-2y = -2x + 2
Divide by -2:
y = x - 1 → slope m₂ = 1

Slopes are equal → parallel

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Problem 2:
Given:
- Line 1: y = -4x + 1 → slope m₁ = -4
- Line 2: 4y = x + 3

Convert Line 2 to slope-intercept:

4y = x + 3
Divide by 4:
y = (1/4)x + 3/4 → slope m₂ = 1/4

Check if perpendicular: m₁ × m₂ = (-4) × (1/4) = -1 → YES!

So, perpendicular

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Problem 3:
Given:
- Line 1: y = (5/6)x - 6 → slope m₁ = 5/6
- Line 2: x + 5y = 4

Convert Line 2:

x + 5y = 4
Subtract x:
5y = -x + 4
Divide by 5:
y = (-1/5)x + 4/5 → slope m₂ = -1/5

Check product: (5/6) × (-1/5) = -5/30 = -1/6 ≠ -1 → not perpendicular
Not equal → not parallel → none

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Problem 4:
Given:
- Line 1: -6x + y = 1
- Line 2: -6x + 3y = -9

Convert both to slope-intercept.

Line 1:
-6x + y = 1 → y = 6x + 1 → slope m₁ = 6

Line 2:
-6x + 3y = -9
Add 6x:
3y = 6x - 9
Divide by 3:
y = 2x - 3 → slope m₂ = 2

Slopes: 6 and 2 → not equal, product = 12 ≠ -1 → none

Wait — let me double-check Line 2:

Original: -6x + 3y = -9
Add 6x: 3y = 6x - 9 → divide by 3 → y = 2x - 3 → yes, slope is 2.

But wait — maybe I made a mistake? Let me check again.

Actually, let’s re-express Line 1 correctly:

Line 1: -6x + y = 1 → y = 6x + 1 → slope 6 ✔️

Line 2: -6x + 3y = -9 → 3y = 6x - 9 → y = 2x - 3 → slope 2 ✔️

Yes, different slopes, not negative reciprocals → none

BUT — hold on! Maybe I misread the second equation?

Looking back at original image: Problem 4 says:

“-6x + y = 1”
“-6x + 3y = -9”

Wait — perhaps we can simplify Line 2 differently?

Let me try dividing entire Line 2 by 3 first:

(-6x)/3 + (3y)/3 = -9/3 → -2x + y = -3 → then y = 2x - 3 → same as before.

Still slope 2.

But what if we compare with Line 1: y = 6x + 1

Nope — still not matching.

Wait — maybe I should check if they’re actually the same line? No, different slopes.

Alternatively — perhaps there's a typo in my reading? Let me look again.

Actually — let me recalculate Line 2 carefully:

Equation: -6x + 3y = -9

Add 6x to both sides: 3y = 6x - 9

Divide by 3: y = 2x - 3 → slope = 2

Line 1: y = 6x + 1 → slope = 6

Product: 6 * 2 = 12 ≠ -1 → not perpendicular
Not equal → not parallel → none

But wait — let me check if maybe the second equation was meant to be something else? Or did I misread?

Looking back at user input: “-6x + 3y = -9” — that’s correct.

Hmm — but let me think — maybe I made an error in Line 1?

Line 1: -6x + y = 1 → add 6x → y = 6x + 1 → correct.

Okay, so answer is none

Wait — no! Wait a minute — let me check if these lines are actually parallel or not by comparing coefficients.

Another way: For two lines A1x + B1y = C1 and A2x + B2y = C2,

They are parallel if A1/A2 = B1/B2 ≠ C1/C2

Perpendicular if A1*A2 + B1*B2 = 0

Let’s use that for Problem 4.

Line 1: -6x + y = 1 → A1=-6, B1=1, C1=1

Line 2: -6x + 3y = -9 → A2=-6, B2=3, C2=-9

Check parallel: A1/A2 = (-6)/(-6) = 1; B1/B2 = 1/3 → 1 ≠ 1/3 → not parallel

Check perpendicular: A1*A2 + B1*B2 = (-6)*(-6) + (1)*(3) = 36 + 3 = 39 ≠ 0 → not perpendicular

So definitely none

Okay, confirmed.

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Now Problems 5–8: Write slope-intercept form of line through given point, parallel to given line.

Remember: Parallel lines have same slope.

Use point-slope form: y - y1 = m(x - x1), then convert to y = mx + b

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Problem 5:
Through (-4, -5), parallel to y = -2x - 5

Slope m = -2 (same as given line)

Point: (x1, y1) = (-4, -5)

Point-slope: y - (-5) = -2(x - (-4))
→ y + 5 = -2(x + 4)

Distribute: y + 5 = -2x - 8

Subtract 5: y = -2x - 13

Final: y = -2x - 13

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Problem 6:
Through (1, 3), parallel to y = x + 5

Slope m = 1

Point: (1, 3)

y - 3 = 1(x - 1)
→ y - 3 = x - 1
Add 3: y = x + 2

Final: y = x + 2

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Problem 7:
Through (5, -5), parallel to y = -(3/2)x - 5

Slope m = -3/2

Point: (5, -5)

y - (-5) = (-3/2)(x - 5)
→ y + 5 = (-3/2)(x - 5)

Distribute right side:
y + 5 = (-3/2)x + (15/2) [because (-3/2)*(-5) = +15/2]

Subtract 5:
y = (-3/2)x + 15/2 - 5
Convert 5 to 10/2:
y = (-3/2)x + 15/2 - 10/2 = (-3/2)x + 5/2

Final: y = -\frac{3}{2}x + \frac{5}{2}

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Problem 8:
Through (-2, 2), parallel to y = (2/3)x - 2

Slope m = 2/3

Point: (-2, 2)

y - 2 = (2/3)(x - (-2))
→ y - 2 = (2/3)(x + 2)

Distribute:
y - 2 = (2/3)x + 4/3

Add 2:
y = (2/3)x + 4/3 + 2
Convert 2 to 6/3:
y = (2/3)x + 4/3 + 6/3 = (2/3)x + 10/3

Final: y = \frac{2}{3}x + \frac{10}{3}

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Now compiling all answers:

1. parallel
2. perpendicular
3. none
4. none
5. y = -2x - 13
6. y = x + 2
7. y = -\frac{3}{2}x + \frac{5}{2}
8. y = \frac{2}{3}x + \frac{10}{3}

Double-checked calculations — all seem correct.

Final Answer:
1. parallel
2. perpendicular
3. none
4. none
5. y = -2x - 13
6. y = x + 2
7. y = -\frac{3}{2}x + \frac{5}{2}
8. y = \frac{2}{3}x + \frac{10}{3}